Interaction estimates and Glimm functional for general hyperbolic systems

We consider the problem of writing Glimm type interaction estimates for the hyperbolic system \begin{equation}\label{E:abs0} u_t + A(u) u_x = 0. \end{equation} %only assuming that $A(u)$ is strictly hyperbolic. The aim of these estimates is to prove that there is Glimm-type functional $Q(u)$ such that \begin{equation}\label{E:abs1} \TV(u) + C_1 Q(u) \ \text{is lower semicontinuous w.r.t.} \ L^1-\text{norm}, \end{equation} with $C_1$ sufficiently large, and $u$ with small BV norm. In the first part we analyze the more general case of quasilinear hyperbolic systems. We show that in general this result is not true if the system is not in conservation form: there are Riemann solvers, identified by selecting an entropic conditions on the jumps, which do not satisfy the Glimm interaction estimate \eqref{E:abs1}. Next we consider hyperbolic systems in conservation form, i.e. $A(u) = Df(u)$. In this case, there is only one entropic Riemann solver, and we prove that this particular Riemann solver satisfies \eqref{E:abs1} for a particular functional $Q$, which we construct explicitly. The main novelty here is that we suppose only the Jacobian matrix $Df(u)$ strictly hyperbolic, without any assumption on the number of inflection points of $f$. These results are achieved by an analysis of the growth of $\TV(u)$ when nonlinear waves of \eqref{E:abs0} interact, and the introduction of a Glimm type functional $Q$, similar but not equivalent to Liu's interaction functional \cite{liu:admis}.